Peel apricots with a vegetable peeler, then halve and pit. Finely chop 2 halves and set aside.
Heat butter in a 12-inch heavy nonstick skillet over medium heat until foam subsides, then cook sugar, stirring constantly, until golden brown. Stir in Amaretto (be careful; syrup will spatter) and simmer, stirring, 2 minutes.
Working in 2 batches, poach apricot halves in syrup at a low simmer, turning, until almost tender, 5 to 10 minutes per batch. Using a slotted spoon, transfer apricots, hollow sides up, to a platter.
Add crumbled amaretti to syrup and cook over low heat, crushing cookies with back of a wooden spoon, until melted into a coarse purée.
Stir in reserved chopped apricot and gently simmer, stirring, until syrup is deep brown and slightly thickened. Cool syrup slightly.
Spoon syrup over apricots and sprinkle with pine nuts (if using). Serve warm or at room temperature.
Cooks' note: Apricots with Amaretto syrup can be made 3 hours ahead and kept, loosely covered, at room temperature.